
theorem
  for C be Simple_closed_curve for n be Nat st n
  is_sufficiently_large_for C holds Y-SpanStart(C,n) < width Gauge(C,n)
proof
  let C be Simple_closed_curve;
  let n be Nat;
A1: len Span(C,n) > 1 by GOBOARD7:34,XXREAL_0:2;
  assume
A2: n is_sufficiently_large_for C;
  then
A3: Span(C,n)/.1 = Gauge(C,n)*(X-SpanStart(C,n),Y-SpanStart(C,n)) by
JORDAN13:def 1;
  [X-SpanStart(C,n),Y-SpanStart(C,n)] in Indices Gauge(C,n) by A2,JORDAN11:8;
  hence thesis by A2,A1,A3,Th25;
end;
